Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings

In this paper, we prove certain one-sided Paley-type inequalities and use them to study the convergence rates for the tail probabilities of sample sums. We then apply our results to find the limiting moments and the limiting distribution of the last time and the largest excess of boundary crossings for the sample sums, generalizing the results previously obtained by Robbins, Siegmund and Wendel. Certain one-sided limit theorems for delayed sums are also obtained and are applied to study the convergence rates of tail probabilities.